where \((\eta _{k})_{k\ge 1}\)  is a sequence of step sizes, which can
be kept constant or decreasing to 0, and \(\xi _{k}\sim {N}(0,\ I_{d})\) 
(\(I_{d}\)  denotes identity matrix dimension \(d\) ), are independent
Gaussian random vectors. It can be seen that Euler-Maruyama discretization,
also known as Langevin Monte Carlo (LMC) algorithm, does not involve
knowledge of \(U\) , but gradient of \(U\)  instead, which makes it ideally
applicable where we typically only know \(U\)  up to a normalizing constant.
Owing to its simplicity, efficiency, and well understood properties,
there are various applications using LMC [1]}, [2]}, [3]}, [4]}, [5]}.
Much of the theory of convergence of sampling used to focus on asymptotic
convergence, failing to provide a detailed study of dimension dependence.
Recently, there is a surge of interests in non-asymptotic rates of
convergence, which include dimension dependence, especially polynomial
dependence on the dimension of target distribution; see, e.g., [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}.
However, there is a critical gap in the theory of discretization of
an underlying stochastic differential equation (SDE) to the broad
spectrum of applications in statistical inference. In particular,
the application of techniques from SDEs traditionally requires \(U(\mathrm {x})\) 
to have Lipschitz-continuous gradients. This requirement prohibits
many typical utilizations [16]}. [17]}
has recently established an original approach to deal with weakly
smooth (possibly non-smooth) potential problems through smoothing.
Their technique rests on results obtained from the optimization community,
perturbing a gradient evaluating point by a Gaussian. However, [17]}
analyzes over-damped Langevin diffusion in the contexts of convex
potential functions while many applications involve sampling in high
dimensional spaces have non convex settings. In another research,
[19]} proposes a very beautiful result using
tail growth for weakly smooth and weakly dissipative potentials. By
using degenerated convex and modified log Sobolev inequality, they
prove that LMC gets \(\epsilon \) -neighborhood of a target distribution
in KL divergence with the convergence rate of \(\tilde{O}(d^{\frac{1}{\alpha }+\frac{1+\alpha }{\alpha }(\frac{2}{\beta }-\mathrm {1}_{\lbrace \beta \ne 1\rbrace })}\epsilon ^{\frac{-1}{\alpha }})\) 
where \(\alpha \)  and \(\beta \)  are degrees of weakly smooth and dissipative
defined in the next section.
